# Arbitrage Theory in Continuous Time

## Synopsis

The second edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. In this substantially extended new edition Bjork has added separate and complete chapters on measure theory, probability theory, Girsanov transformations, LIBOR and swap market models, and martingale representations, providing two full treatments of arbitrage pricing: the classical delta-hedgingand the modern martingales. More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs.

## Excerpt

The purpose of this book is to present arbitrage theory and its applications to pricing problems for financial derivatives. It is intended as a textbook for graduate and advanced undergraduate students in finance, economics, mathematics, and statistics and I also hope that it will be useful for practitioners.

Because of its intended audience, the book does not presuppose any previous knowledge of abstract measure theory. The only mathematical prerequisites are advanced calculus and a basic course in probability theory. No previous knowledge in economics or finance is assumed.

The book starts by contradicting its own title, in the sense that the second chapter is devoted to the binomial model. After that, the theory is exclusively developed in continuous time.

The main mathematical tool used in the book is the theory of stochastic differential equations (SDEs), and instead of going into the technical details concerning the foundations of that theory I have focused on applications. The object is to give the reader, as quickly and painlessly as possible, a solid working knowledge of the powerful mathematical tool known as Itô calculus. We treat basic SDE techniques, including Feynman-Kač representations and the Kolmogorov equations. Martingales are introduced at an early stage. Throughout the book there is a strong emphasis on concrete computations, and the exercises at the end of each chapter constitute an integral part of the text.

The mathematics developed in the first part of the book is then applied to arbitrage pricing of financial derivatives. We cover the basic Black-Scholes theory, including delta hedging and “the greeks”, and we extend it to the case of several underlying assets (including stochastic interest rates) as well as to dividend paying assets. Barrier options, as well as currency and quanto products, are given separate chapters. We also consider, in some detail, incomplete markets.

American contracts are treated only in passing. The reason for this is that the theory is complicated and that few analytical results are available. Instead I have included a chapter on stochastic optimal control and its applications to optimal portfolio selection.

Interest rate theory constitutes a large part of the book, and we cover the basic short rate theory, including inversion of the yield curve and affine term structures. The Heath-Jarrow-Morton theory is treated, both under the objective measure and under a martingale measure, and we also present the Musiela parametrization. The basic framework for most chapters is that of a multifactor model, and this allows us, despite the fact that we do not formally use measure

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